An initial-boundary value problem for the Korteweg-de Vries equation on the negative quarter-plane

uτ+uux+uxxx=0,x<0,τ>0,where x and τ represent dimensionless distance and time, respectively. In particular, we consider the case when the initial and boundary conditions are given by u(x,0)=ui for x<0 and u(0,τ)=ub,∂∂xu(0,τ)=ubx for τ>0. Here the initial value ui<0 and we restrict attention to boundary values ub and ubx in the ranges 013(ub-ui)(-ub-2ui)12. We also present detailed numerical simulations of the full initial-boundary value problem which support the asymptotic analysis presented. A brief discussion is also given of the large-τ asymptotic structure to this problem when ui<0,ub⩾-2ui and ubx∈(-∞,∞).